30. Describe a circle of given radius to pass through a given point, and touch a given straight line. 31. Describe a circle of given radius to touch two given circles. 32. Describe a circle of given radius to touch two given straight lines. 33. Describe a circle of given radius to touch a given circle and a given straight line. 34. Describe two circles of given radii to touch one another and a given straight line, on the same side of it. 35. If a circle touches a given circle and a given straight line, shew that the points of contact and an extremity of the diameter of the given circle at right angles to the given line are collinear. 36. To describe a circle to touch a given circle, and also to touch a given straight line at a given point. Let DEB be the given circle, PQ the given st. line, and A the given point in it: it is required to describe a circle to touch the DEB, and also to touch PQ at A. At A draw AF perp. to PQ: 1. 11. then the centre of the required circle must lie in AF. III. 19. Find C, the centre of the DEB, Ε B III. 1. and draw a diameter BD perp. to PQ: join A to one extremity D, cutting the Oce at E. A 77 Join CE, and produce it to cut AF at F. Then F is the centre, and FA the radius of the required circle. [Supply the proof: and shew that a second solution is obtained by joining AB, and producing it to meet the Oce also distinguish between the nature of the contact of the circles, when PQ cuts, touches, or is without the given circle.] 37. Describe a circle to touch a given straight line, and to touch a given circle at a given point. 38. Describe a circle to touch a given circle, have its centre in a given straight line, and pass through a given point in that straight line. [For other problems of the same class see page 235.] ORTHOGONAL CIRCLES. DEFINITION. Circles which intersect at a point, so that the two tangents at that point are at right angles to one another, are said to be orthogonal, or to cut one another orthogonally. 39. In two intersecting circles the angle between the tangents at one point of intersection is equal to the angle between the tangents at the other. 40. If two circles cut one another orthogonally, the tangent to each circle at a point of intersection will pass through the centre of the other circle.. 41. If two circles cut one another orthogonally, the square on the distance between their centres is equal to the sum of the squares on their radii. 42. Find the locus of the centres of all circles which cut a given circle orthogonally at a given point. 43. Describe a circle to pass through a given point and cut a given circle orthogonally at a given point. III. ON ANGLES IN SEGMENTS, AND ANGLES AT THE See Propositions 20, 21, 22; 26, 27, 28, 29; 31, 32, 33, 34. 1. If two chords intersect within a circle, they form an angle equal to that at the centre, subtended by half the sum of the arcs they cut off. Let AB and CD be two chords, intersecting at E within the given ADBC: then shall the AEC be equal to the angle at the centre, subtended by half the sum of the arcs AC, BD. Join AD. Then the ext. 4 AEC the sum of the int. E B opp. SEDA, EAD; that is, the sum of the CDA, BAD. But the CDA, BAD are the angles at A the Oce subtended by the arcs AC, BD; .. their sum = half the sum of the angles at the centre subtended by the same arcs; or, the AEC=the angle at the centre subtended by half the sum of the arcs AC, BD. Q. E. D. 2. If two chords when produced intersect outside a circle, they form an angle equal to that at the centre subtended by half the difference of the arcs they cut off. 3. The sum of the arcs cut off by two chords of a circle at right angles to one another is equal to the semi-circumference. 4. AB, AC are any two chords of a circle; and P, Q are the middle points of the minor arcs cut off by them: if PQ is joined, cutting AB and AC at X, Y, shew that AX=AY. 5. If one side of a quadrilateral inscribed in a circle is produced, the exterior angle is equal to the opposite interior angle. 6. If two circles intersect, and any straight lines are drawn, one through each point of section, terminated by the circumferences; shew that the chords which join their extremities towards the same parts are parallel. 7. ABCD is a quadrilateral inscribed in a circle; and the opposite sides AB, DC are produced to meet at P, and CB, DA to meet at Q: if the circles circumscribed about the triangles PBC, QAB intersect at R, shew that the points P, R, Q are collinear. 8. If a circle is described on one of the sides of a right-angled triangle, then the tangent drawn to it at the point where it cuts the hypotenuse bisects the other side. 9. Given three points not in the same straight line: shew how to find any number of points on the circle which passes through them, without finding the centre. 10. Through any one of three given points not in the same straight line, draw a tangent to the circle which passes through them, without finding the centre. 11. Of two circles which intersect at A and B, the circumference of one passes through the centre of the other from A any straight. line is drawn to cut the first at C, the second at D; shew that CB= CD. 12. Two tangents AP, AQ are drawn to a circle, and B is the middle point of the arc PQ, convex to A. Shew that PB bisects the angle APQ. 13. Two circles intersect at A and B; and at A tangents are drawn, one to each circle, to meet the circumferences at C and D : if CB, BD are joined, shew that the triangles ABC, DBA are equiangular to one another. 14. Two segments of circles are described on the same chord and on the same side of it; the extremities of the common chord are joined to any point on the arc of the exterior segment: shew that the arc intercepted on the interior segment is constant. 15. If a series of triangles are drawn standing on a fixed base, and having a given vertical angle, shew that the bisectors of the verti cal angles all pass through a fixed point. 16. ABC is a triangle inscribed in a circle, and E the middle point of the arc subtended by BC on the side remote from A: if through E a diameter ED is drawn, shew that the angle DEA is half the difference of the angles at B and C. [See Ex. 7, p. 101.] 17. If two circles touch each other internally at a point A, any chord of the exterior circle which touches the interior is divided at its point of contact into segments which subtend equal angles at A. 18. If two circles touch one another internally, and a straight line is drawn to cut them, the segments of it intercepted between the circumferences subtend equal angles at the point of contact. 19. THE ORTHOCENTRE OF A TRIANGLE. The perpendiculars drawn from the vertices of a triangle to the opposite sides are concurrent. In the AABC, let AD, BE be the perps drawn from A and B to the opposite sides; and let them intersect at O. Join CO; and produce it to meet AB at F. It is required to shew that CF is perp. .. the points O, E, C, D are concyclic : .. the DEC=the DOC, in the same segment; Again, because the AEB, ADB are rt. angles, .. the points A, E, D, B are concyclic: .. the DEB = the DAB, in the same segment. 8 .. the sum of the 4 FOA, FAO = the sum of the 4 DEC, =a rt. angle: AFO-a rt. angle: .. the remaining Hence the three perps AD, BE, CF meet at the point O. [For an Alternative Proof see page 106.] O D E Hyp. DEB Hyp. 1. 32. Q. E. D. DEFINITIONS. (i) The intersection of the perpendiculars drawn from the vertices of a triangle to the opposite sides is called its orthocentre. (ii) The triangle formed by joining the feet of the perpendiculars is called the pedal or orthocentric triangle. 20. In an acute-angled triangle the perpendiculars drawn from the vertices to the opposite sides bisect the angles of the pedal triangle through which they pass. In the acute-angled ▲ ABC, let AD, BE, CF be the perps drawn from the vertices to the opposite sides, meeting at the orthocentre O; and let DEF be the pedal triangle: then shall AD, BE, CF bisect respectively the FDE, DEF, EFD. For, as in the last theorem, it may B be shewn that the points O, D, C, E are concyclic; .. the ODE= the OCE, in the same segment. Similarly the points O, D, B, F are concyclic; .. the ODF the OBF, in the same segment. E But the OCE=the OBF, each being the compt of the ▲ BAC. ..the ODE=the Z ODF. Similarly it may be shewn that the BE and CF. DEF, EFD are bisected by Q. E. D. COROLLARY. (i) Every two sides of the pedal triangle are equally inclined to that side of the original triangle in which they meet. FDB = the BAC, B, Similarly it may be shewn that the .. the EDC=the FDB In like manner it may be proved that the DEC=the FEA= the and the DFB the EFA the ▲ C. COROLLARY. (ii) The triangles DEC, AEF, DBF are equiangular to one another and to the triangle ABC.· NOTE. If the angle BAC is obtuse, then the perpendiculars BE, CF bisect externally the corresponding angles of the pedal triangle. |